The finite element solution of problems involving three-dimensional acoustic waves in an infinite wave guide, and in the infinite medium around a structure is considered. Such problems are typical in structural acoustics, and this paper concentrates on the efficient numerical treatment of the infinite acoustic medium away from the structure. The unbounded domain is truncated by means of an artificial boundary B. On B, non-reflecting boundary conditions are used; these are either nonlocal Dirichlet-to-Neumann conditions, or their localized counterparts. For the high-order localized conditions, special three-dimensional finite elements are constructed for use in the layer adjacent to B. The performance of the nonlocal and localized boundary conditions is compared via numerical experiments involving a three-dimensional wave guide.

The Hagstrom–Warburton high-order absorbing boundary conditions (ABCs) are considered. They are based on a high-order form of the Higdon ABCs using auxiliary variables and constitute a modification of the previously proposed Givoli–Neta ABCs. Here the Hagstrom–Warburton ABCs, which were originally used in a finite difference scheme, are incorporated into a finite element formulation. Exterior time-dependent problems are considered with rectangular computational domains. Special corner conditions are used in conjunction with the ABCs to make the truncated problem well-posed. The properties of the Hagstrom–Warburton and Givoli–Neta formulations are compared, and the relations between the two formulations are established. Numerical examples demonstrate the performance of the Hagstrom–Warburton finite element scheme.

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ℬ︁, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ℬ︁. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C 0 semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ℬ︁. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.

The reduction of the large in-plane static deformation of a thin hyperelastic plate using control loads is considered. This problem has important applications in the control of flexible space structures. A mathematical model leads to an elliptic optimal control problem in nonlinear elasticity. A numerical optimal control method, based on Finite Element (FE) discretization and Sequential Quadratic Programming (SQP), is employed to minimize the deformation of the plate. Results are presented for a specific example.

One of the methods commonly used to numerically solve a problem in an infinite domain is the method of artificial boundary conditions [1]. For a linear scalar problem, this method may be summarized as follows:

A finite element method for the solution of linear elliptic problems in infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-infinite strip, are considered in detail. In the proposed method, an artificial boundary B is first introduced, to make the computational domain Omega finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B. This condition is localized, and a sequence of local boundary conditions on B, of increasing order, is obtained. The problem in Omega, with a localized DtN boundary condition on B, is then solved using the finite element method. The numerical stability of the scheme is discussed. A hierarchy of special conforming finite elements is developed and used in the layer adjacent to B, in conjunction with the local high-order boundary condition applied on B. An error analysis is given for both nonlocal and local boundary conditions. Numerical experiments are presented to demonstrate the performance of the method.